Introduction E;Cghrist/preprints/pursuit...Theorem 4.1. For coverage pair (E;C) over Rq with dimE>2 and 0 6q6dimE 2, +HdimE q 1C˘=+H q(E C) Along the way, we investigate properties

POSITIVE ALEXANDER DUALITY FOR PURSUIT AND EVASION

ROBERT GHRIST AND SANJEEVI KRISHNAN

Abstract. Considered is a class of pursuit-evasion games, in which an evader tries toavoid detection. Such games can be formulated as the search for sections to the com-

plement of a coverage region in a Euclidean space over a timeline. Prior results givehomological criteria for evasion in the general case that are not necessary and sufficient.

This paper provides a necessary and sufficient positive cohomological criterion for evasion

in a general case. The principal tools are (1) a refinement of the Cech cohomology of acoverage region with a positive cone encoding spatial orientation, (2) a refinement of the

Borel-Moore homology of the coverage gaps with a positive cone encoding time orienta-

tion, and (3) a positive variant of Alexander Duality. Positive cohomology decomposesas the global sections of a sheaf of local positive cohomology over the time axis; we show

how this decomposition makes positive cohomology computable as a linear program.

1. Introduction

The motivation for this paper comes from a type of pursuit-evasion game. In such games,two classes of agents, pursuers and evaders move in a fixed geometric domain over time.The goal of a pursuer is to capture an evader (by, e.g., physical proximity or line-of-sight).The goal of an evader is to move in such a manner so as to avoid capture by any pursuer.This paper solves a feasibility problem of whether an evader can win in a particular settingunder certain constraints.

We specialize to the setting of pursuers-as-sensors, in which, at each time, a certainregion of space is “sensed” and any evader in this region is considered captured. Evasion,the successful evasion of all pursuers by an evader, corresponds to gaps in the sensed regionover time. We formalize this setting in the form of a coverage pair (E,C) over the time-axisR: one has, for each time t, the ambient space Et ∼= Rn and a sensed or covered regionCt ⊂ Et. The ambient space and covered regions at each point in time are bundled into anambient space-time E ∼= Rn × R and a covered subspace C ⊂ E. The goal of the evader isto construct an evasion path, a section to the restriction E−C → R of the projection mapE → R.

This is not necessarily a topological problem. However, epistemic restrictions on C arenatural in real-world settings. For example, a distributed sensor network may not directlyperceive its geometric coverage region C but can often infer its (co)homology from localcomputations. The present paper focuses on the feasibility of evasion:

Problem. Is there an “evasion criterion” for the existence of a section to the complement

E−C → R,

based on a coordinate-free description of C?

1991 Mathematics Subject Classification. Primary:55N30, 55S40, 55U30 Secondary: 57R19, 90C48, 91A44.Authors supported by US DoD contracts FA9550-12-1-0416 and N00014-16-1-2010.

1

2 ROBERT GHRIST AND SANJEEVI KRISHNAN

Work of Adams and Carlsson gives a geometric criterion for the plane (n = 2, see below);there is no other known sharp criterion. The problem is one of topological duality. Whilethe (timewise) unstable homotopy theory of E−C determines the existence of an evasionpath, currently only the less informative (timewise) stable homotopy theory of E−C admitsa duality with the (timewise) stable homotopy theory of the input space C.

1.1. Background. Different formulations of pursuit-evasion games suggest different meth-ods of analysis. Combinatorial descriptions of the domain suggest methods from the theoryof graphs [18] and cellular automata [4, 6]. Geometric descriptions of agent behavior suggestmethods from differential equations and differential game theory [15], computational geome-try [13, 19], probability theory [14, 21], and Alexandrov geometry [2, 3]. Coordinate-free de-scriptions of pursuer behavior model the data available to ad-hoc networks of non-localizedsensors, to which coordinate geometry is unavailable; such descriptions suggest methodsfrom topology [1, 9, 10].

The first topological criterion for evasion is based on the homologies of the fibers Ctrelative their topological boundaries in Et [9]. Assume for convenience that coverage gapsare uniformly bounded in the sense that there is a sufficiently large, closed ball B ⊂ Rnsuch that Rn−B ⊂ Ct at each time t ∈ R. The topological boundary F of this ball B inRn encircles all potential losses of coverage like a fence; this fence naturally determines ahomology class [F ] in Hn−1C. The homological criterion says that if there is an evasionpath, then [F ] 6= 0. This criterion for evasion, while necessary, is not sufficient [Figure 1].

Figure 1. Three coverage pairs over R, all planar examples (n = 2). Eva-sion is possible in the first case only [left]. The simple homological criterionrules out evasion in the second [middle], but not third [right], case.

Another topological criterion for evasion follows from Alexander Duality (classical orparametrized [16]), a duality between the cohomology of the fiberwise coverage regions Ctand the homology of the complementary coverage gaps Et−Ct [1]. One such criterion statesthat if there is an evasion path, then there exists a persistent barcode in the zigzag persistenthomology [7] of C. This criterion for evasion, while necessary, is not sufficient [Figure 1,right]. The main problem, as observed in [1], is that the fiberwise homotopy type of C failsto detect enough information about the embedding C ↪→ E, and hence E−C, to deduceevasion: see Figures 5 and 6.

A necessary and sufficient refinement of the previous criterion exists for the special casewhere n = 2 and Ct is a connected union of balls at all times t that only changes topologyat finitely many points t in time [1]. In this case, an algorithm iteratively constructs anevasion path at each time t of topology change (via a Reeb graph) based on the affine andorientation structures Ct inherits from Rn. An open question in that paper is whether thereis a generalization that does not require information about affine structure.

POSITIVE ALEXANDER DUALITY FOR PURSUIT AND EVASION 3

1.2. Contributions. We give a necessary, sufficient, and tractable cohomological criterionfor evasion that affirmatively answers this open question in a general case and arbitrarydimension. The (co)homology theories of parametrized spaces used in this paper, at least inthe cases considered in the main results, are equivalent to real Borel-Moore homology havingproper supports and real Cech cohomology. A positive cone +HqP inside the homology HqPof a space P over Rq, reminiscent and sometimes a special case of cones in the homology ofa directed space [12], encodes some information about the parametrization. A positive cone+HkP inside the cohomology HkP of a space P over Rq, whose fibers are endowed withthe additional structure of pro-objects in a category of oriented k-manifolds, encodes someinformation about those orientations. The main result is the following positive variant ofAlexander Duality.

Theorem 4.1. For coverage pair (E,C) over Rq with dimE > 2 and 0 6 q 6 dimE − 2,

+HdimE−q−1C ∼= +Hq(E−C)

Along the way, we investigate properties of positive (co)homology. We show that pos-itive cohomology, defined on fiberwise oriented manifolds-with-compact-boundaries, sendscofiltered limits to colimits, hence is well-defined on pro-objects in a category of orientedmanifolds-with-compact boundaries, and in particular extends to closed subspaces of fiber-wise oriented manifolds whose complements have proper closures. We show that positivehomology: (1) preserves certain limits [Lemma 2.7] and (2) characterizes the existence ofsections [Lemmas 2.8, 2.9] in degree 1. Consequently, we conclude a complete criterion forevasion by considering Positive Alexander Duality in homological degree q = 1.

Corollary 4.2. For a coverage pair (E,C) over R with dimE > 2,

(1) +HdimE−2C 6= ∅

if and only if the restriction E−C → R of the projection E → R admits a section.

This complete criterion for evasion assumes a minimum of information about the coverageregion. Unlike [1], we neither assume that our coverage region has the homotopy typeof a CW complex, nor restrict the ambient spatial dimension of the region to dimEt =dimE−1 = 2, nor restrict the manner in which the topology of our coverage region changesin time. In addition, the criterion is only based on the Cech cohomology and orientationdata of the coverage region.

Moreover, this evasion criterion is efficiently computable. The positive cohomology ofa coverage region over R decomposes as the global sections of a sheaf of local positivecohomology cones.

Corollary 5.2. For coverage pair (E,C) over R and collection O of open subsets of R,

+HdimE−2C = limU∈O

+HdimE−2CU .

Abstractly, the global sections of a cellular sheaf of finitely generated, positive cones overa finitely stratified space is computable as a linear programming problem. We can thereforedemonstrate the tractability of the evasion criterion with a prototypical pair of examples in§6.

We use the language of sheaf theory throughout, as part of a broader goal in introduc-ing sheaf-theoretic methods in applied and computational topology. Below we fix someconventions in notation and terminology regarding sheaves, spaces, and cones.


1.2.1. Topology. Let X denote a space. We write clo(X) and com(X) for the respective setsof closed and compact subsets of X. We write π0X for the set of path-components naturalin X. For B ⊂ X and a collection Ψ of subsets of X, we write Ψ|B and Ψ ∩B for

Ψ|B = {A ∈ Ψ | A ⊂ B}, Ψ ∩B = {A ∩B | A ∈ Ψ}

A manifold-with-compact-boundary is a manifold-with-boundary whose boundary is com-pact. Let Mn denote the category of non-compact, connected, oriented manifolds-with-compact boundary and oriented embeddings ψ : M → N between such oriented manifoldswith ψ(M) ⊂ N−∂N .

A space over X is a space P equipped with a map τP : P → X. A section to a space Pover X is a section to τP . For in inclusion ι : U ↪→ V of spaces and a space P over V , wewrite PU for the space over U defined by the pullback square

PU //

τPU

��J

P

τP

��U

ι// V

A pair of spaces over X is a pair (P,Q) of spaces over X with Q a subspace of P and τQ thepullback of τP along inclusion Q ↪→ P . A space P over X is proper if τP is proper. A cubecomplex over Rq is a locally finite union P of isothetic (axis-aligned) compact hyperrectanglesin Rn+q with τP projection onto the last q coordinates. For a space P over X, let secP denotethe sheaf on X sending each open U ⊂ X to the set of sections to PU .

1.2.2. Sheaves. Fix a space X and a collection Ψ of closed subsets of X closed under inter-section. Let ConstX;V denote the constant sheaf on X taking values in an object V . LetF(A) denote the pullback of a sheaf F on X to A ⊂ X. Let ΓF denote the global sectionsof a sheaf F .

We fix some constructions and notation for Abelian sheaf theory; refer to [5] for details.Consider a sheaf F of real vector spaces on X. Let ΓΨF denote the global sections of Fwhose supports lie in the collection Ψ of subsets of the base space. For an injective resolutionI• of F , we write HΨ

• and H•Ψ for

HΨ• F = H•ΓΨ(U 7→ (Γcom|UI•)∗)

H•ΨF = H•ΓΨI•,

For simplicity, we do not define relative sheaf (co)homology and instead restrict ourattention to pairs (X,A) on which the relative theory reduces to an absolute theory. Fixclosed A ⊂ X and suppose Ψ is a collection of paracompact subsets of X which each admita neighborhood in X which is an element in Ψ. There exists a long exact sequence

(2) · · · → H•ΨF → H•Ψ∩AF(A)∂•−→ H•+1

Ψ|X−AF(X−A) = H•+1Ψ|X−AF → · · · ,

[5, Proposition 12.3]. If additionally X has finite cohomological dimension (in the sensethat the compactly supported cohomology of open subsets of X vanishes for large enoughdegrees [5]), then there exists a long exact sequence

(3) · · · → HΨ• F → H

Ψ∩(X−A)• F(X−A) = H

Ψ∩(X−A)• F ∂•−1−−−→ H

Ψ|A•−1F(A) → · · ·

[5, (95),§13.2]. In both exact sequences, the unlabelled arrows are evident inclusions andrestrictions.


For X an oriented n-manifold in Ψ, we write ∆ for Poincare Duality

∆ : Hn−pΨ F = HΨ

p F , q = 0, 1, . . . , n

1.2.3. Cones. In this paper, a cone is a subset of a real vector space V closed under vectoraddition and closed under scalar multiplication by R+, the positive real numbers. A coneK in V is positive if K ∩ −K = ∅. Let Cones denote the category whose objects are conesand whose morphisms are functions between cones that are restrictions and corestrictionsof linear maps between vector spaces generated by the cones.

Example 1.1. For real vector spaces V and W ,

Cones(V,W )

is the set of linear maps V → W and hence can be regarded itself as a real vector spacewith vector addition and scalar multiplication defined point-wise.

For cones V +,W+ in real vector spaces V,W with V generated by V +, we regard

(4) Cones(V +,W+) ⊂ Cones(V,W )

as a cone inside the vector space Cones(V,W ) under the injection Cones(V +,W+) →Cones(V,W ) sending a cone map to its unique linear extension.

1.3. The Formal Problem. The setting for pursuit-evasion in this paper is a coveragepair.

Definition 1.2. A coverage pair over a space T is a pair (E,C) of spaces over T such that:

(1) E is a real vector bundle over T ; and(2) C is a closed and connected subspace of E; and(3) The closure of E−C in E is proper.

We make some immediate remarks on the definition. Firstly, we stress the connectednessimplicit in the definition of a coverage pair: as noted in [1], this is a necessary condition forevasion criteria. Secondly, the ambient space E of a coverage pair (E,C) over a contractiblespace T is homeomorphic to Rn×T ; for the case T = Rq, we always take E to be an orientedmanifold. Thirdly, the motivating example of T is T = R — the setting for temporal evasionproblems. To the extent possible, we develop our machinery for more general base spaceswith an eye towards future uses in evasion-type problems over directed or partially-directedparameter spaces or state spaces.

We wish to regard the fibers of C as oriented. Hence we generalize orientations for generalspaces as the structure of pro-objects in a category of oriented manifolds. We identify eachfiber Ct of a coverage region C in a coverage pair (E,C) over T with the pro-object inMdimEt

represented by the inverse system of closed neighborhoods of Ct in Et which areobjects in MdimEt

.

2. Homology

In this section, let P denote a space with finite cohomological dimension over anotherspace such that each proper subspace of P is paracompact and admits a proper neighborhoodin P . A motivating example of P is the complement E−C of the coverage region in theambient space of a coverage pair (E,C) over Euclidean space. For each such P , let prop(P )denote the collection of proper subspaces of P and define

H•P = Hprop(P )• (ConstP ;R).

(This variant of homology differ subtly from the usual singular and Borel-Moore theories.)


Example 2.1. For a cube complex P over R and real numbers x < y,

H•P(x,y) = Hsing• (P[x,y], P{x, y}),

where Hsing• denotes real singular homology.

The construction H• is contravariant in maps of the form PU ↪→ P , for U an opensubspace of the codomain of τP , and covariant in maps of the form PA ↪→ P , for A a closedsubspace of the codomain of τP . For U an open subspace of the space to which P maps,the long exact sequence (3) specializes to the following long exact sequence, in which theunlabelled arrows are induced from inclusions and restrictions.

· · · → HkPU∂k−→ Hk−1(P−PU )

Hk−1(P−PU ↪→P )−−−−−−−−−−−→ Hk−1P → · · ·

+

+

Figure 2. Non-Positive Homology Class The path at the core of thesolid tube P over R represents an element in H1P that does not lie in thepositive cone +H1P because the path intersects a fiber negatively.

2.1. Positive homology definition. Our goal is to define a positive homology that re-members directedness: see Figure 2. To that end, we observe the following convolutedreinterpretation of ordinary H0 of a space X over a point:

H0X = colimcom K⊂XCones(Γ(ConstK;R),R).

This points to a simple modification for definition positive homology in degree zero:

Definition 2.2. Define +H0P to be the cone

+H0P = colimcom K⊂PCones(Γ(ConstK;R+),R+)

natural in spaces P (over R0).

Recall that we can regard +H0P to be a cone inside H0P by (4).

Example 2.3. For a CW complex P over a point, +H0P is the positive cone generated by

π0P ⊂ R[π0P ] = H0P.

and hence remembers unstable homotopical information about P inside the stable homo-topical invariant H0P .

We may then define positive homology in arbitrary degree over more general base spaces(open subsets of a directed Rq) inductively, using pullback squares. Given a space P overan open U ⊂ Rq, we write Pt for the space over Rq−1 such that τPt

is the composite

P{t}×Rq → {t} × Rq−1 → Rq−1


of a restriction and corestriction of τP with projection {t} × Rq−1 → Rq−1 onto the lastq − 1 factors and P(−∞,t) for PU∩((−∞,t)×Rq−1).

Definition 2.4. For positive integers q, define +HqP by the pullback square

(5) +HqP //

��J

∏t

+Hq−1Pt

��HqP ∏

tHqPHq(P(−∞,t)⊂P )

−−−−−−−−−−→HqP(−∞,t)

∂q−→Hq−1Pt

// ∏tHq−1Pt,

where t denotes an element in R such that {t} × Rq−1 ∩ U 6= ∅, natural in spaces P overopen U ⊂ Rq, in Cones.

Example 2.5. We are particularly interested in +H1. In this case, (5) is easy to interpret forP an open subspace of a vector bundle over R: a positive homology class is represented by alocally finite singular 1-cycle, whose 1-chains are oriented in the sense that their compositeswith τP are non-decreasing maps I→ R. Figure 3 illustrates four examples of +H1 of a 1-dspace over R.

(a)

(b)

(c)

(d)

Figure 3. Positive 1-homology cones: Four simple examples of 1-dspaces over R. (a) H1

∼= R and +H1∼= R+. (b) Here, also, H1

∼= R, but,as there is a reversal in the path, +H1 = ∅. (c) With the wedged circle,H1∼= R × R and +H1

∼= R+ × R. (d) In this last example, there are fourdistinct sections over the base space and +H1 is a cone with four vectorsas a basis. However, H1

∼= R3, since only three are linearly independent.Thus +H1 is a cone in R3 generated by four vectors.

Example 2.6. For the space B over Rq such that τB = 1Rq , +HqB = R+.

A homology cone is always positive. Hence we call +H•P the positive homology of P .Positive homology inherits from H• a covariance in closed inclusions of spaces over R andcontravariance in inclusions of the form PU ↪→ PV for spaces P over R and open U ⊂ V ⊂ R.


2.2. Positive homology limits and colimits. In the most relevant case of degree onehomology of spaces over R, +H1 defines a sheaf over R.

Lemma 2.7. For a space P over R and collection O of open subsets of R,+H1P = lim

U∈O

+H1PU

Proof. Let +H0,H0,+H1,H1 be the presheaves on R naturally defined by

+H0(U) =∏x∈U

+H0Px,+H1(U) = +H1PU ,

H0(U) =∏x∈U

H0Px, H1(U) = H1PU .

Let ∂ denote the sheaf map H1 → H0 such that

∂U =∏x∈R

(H1PU∩(−∞,x)∂1−→ H0Px) ◦H1(PU∩(−∞,x) ⊂ P ).

There exists an isomorphism of presheaves

(6) H1 ×∂ +H0∼= +H1.

The presheaves +H0 and H0 are sheaves because products commute with limits. Thepresheaf H1 is a sheaf by an application of the First Fundamental Theorem of Sheaves [5,Theorem IV.2.1]. Hence the left side of (6), and hence also the right side of (6), is a sheafbecause section-wise limits of sheaves are sheaves. �

2.3. Detecting sections. Sections to a space P over Rq determine positive elements inHqP .

Lemma 2.8. For a space P over Rq admitting a section, +HqP 6= ∅.

Proof. A section to P defines a closed inclusion B → P of spaces over Rq, with B the spaceover Rq defined by τB = 1Rq , and hence induces a cone map +HqB → +HqP with domainisomorphic to R+ and hence non-empty. �

A converse holds for homological degree 1, under some point-set conditions.

Lemma 2.9. An open subspace P of a vector bundle over R admits a section if +H1P 6= ∅.

Proof. We take P to be an open subspace of the bundle Rn+1 over R, whose bundle map isprojection onto the last factor, without loss of generality by the triviality of vector bundlesover R. Let R+[−] denote the left adjoint of the forgetful functor from Cones to the categoryof sets and functions.

Suppose +H1P 6= ∅. There exists a proper cubical complex K ⊂ P over R with +H1K 6=∅. We can endow R with the structure of a CW complex such that Ke is the product of acompact hyperrectangle with e, for each open edge e ⊂ R, by K a cubical complex. Let σtbe the composite

H1KH1(K(−∞,t)⊂K)−−−−−−−−−−−→ H1K(−∞,t)

∂1−→ H0Kt

We can define ηc : H1K → R[π0secK(c)] for each cell c in R by the commutative squares

+H0Kv R+[π0Kv]

+H1K

σv

OO

ηv// R+[π0secK(v)],

R+[π0(s7→s(v))]

OO+H0Kv+w/2 R+[π0Kv+w/2]

+H1K

σv+w/2

OO

η[v,w]

// R+[π0secK([v, w])]

R[π0(s7→s(v+w/2))]

OO


in which v < w denote successive vertices in R, because the right vertical arrows are in-vertible - in the left square by s 7→ s(v) : secK(v) ∼= Kv and in the right square byour choice of CW structure on R. It is straightforward to verify that η defines a coneη : +H1K → R+[π0secK ]. Hence the limit limc π0secK(c) over all cells of R is non-empty,and hence ΓsecK 6= ∅ [Lemma A.1]. �

3. Cohomology

For each paracompact space X, take H•X to mean the sheaf cohomology

H•X = H•clo(X)(ConstX;R).

Example 3.1. The construction H• is equivalently real Cech cohomology.

The construction H•X is contravariant in spaces X. For a paracompact space X andclosed A ⊂ X, the long exact sequence (2) specializes to the following long exact sequence,in which the unlabelled arrows are induced from inclusions and restrictions.

(7) · · · → H•X → H•(X−A)∂•−→ H•+1A→ · · · ,

For each cofiltered system Xi of paracompact spaces and embeddings between them,

colimiHn−1Xi = Hn−1 lim iXi

where the limit above is taken in the category of spaces and maps between them [5, Theorem10.6].

Positive cones on the cohomologies of oriented manifolds-with-compact-boundary encodeinformation about those orientations as follows.

Definition 3.2. Define +Hn−1M by the pullback square

+Hn−1M //

��J

+H0∂M

��Hn−1M

∆◦Hn−1(∂M⊂M)

// H0∂M,

where the right vertical arrow is inclusion, natural in non-compact, connected, orientedn-manifolds-with-compact-boundary.

It is straightforward to check that +Hn−1 is contravariant in morphisms in Mn. Moreover,+Hn−1 sends cofiltered limits in Mn to colimits in Cones. Thus the following extension of+Hn−1 is well-defined.

Definition 3.3. Define +Hn−1M for a pro-object M in Mn by

+Hn−1M = colimi+Hn−1Mi.

We extend the definition of positive cohomology one final time as follows.

Definition 3.4. Define +Hn−1P by the pullback square

+Hn−1P //

��J

∏x∈T

+Hn−1Px

��Hn−1P ∏

x∈T Hn−1(Px⊂P )

// ∏x∈T H

n−1Px,


where the right vertical arrow is inclusion, natural in spaces P over T whose fibers have theextra structure of decompositions, in the category of spaces, as limits of pro-objects in Mn.

The cone +Hn−1P inside Hn−1 limi Pi is always positive. Hence we call +Hn−1P thepositive cohomology of P .

Figure 4. Cohomological counterparts to +H0,+H1: In the manifold-

with-boundary M , left, a compactly supported 1-cohomology class is posi-tive if and only if its restriction to the boundary is a positive multiple of theorientation class of that boundary. In the space P over R shown [middle],a 1-cohomology class is positive if its restriction to each fiber is positive[right].

4. Alexander Duality

Alexander Duality AD is an isomorphism defined by the commutative diagram

HdimE−q−1CAD //

∂n−1

��

Hq(E−C)

HdimE−q(E−C)∆

// Hq(E−C)

of isomorphisms. Alexander Duality restricts and corestricts to an isomorphism of positivecones.

Theorem 4.1. For coverage pair (E,C) over Rq with dimE > 2 and 0 6 q 6 dimE − 2,

+HdimE−q−1C ∼= +Hq(E−C)

along a restriction and corestriction of Alexander Duality.

Proof. We induct on q.Consider the base case q = 0. It suffices to consider the subcase q = and C is an

oriented dimEt-submanifold-with-compact-boundary. The more general case q = 0 followsfrom taking the colimit, over neighborhoods M of C in E that fall under the aforemntioned


special case, of positive Alexander Dualities of the form +HdimE−1M ∼= +H0(E−M) bydefinition of +HdimE−1C and H0 compactly supported.

Consider the following commutative diagram of solid arrows, in which the top left squareis a pullback square and ιX denotes inclusion +H0X ↪→ H0X.

(8) +HdimE−1Cα+

//

��

+H0∂Cβ+

//

∆−1◦ι∂C

��

+H0(E−C)

ιE−C

��HdimE−1C

HdimE−1(∂C⊂C)

//

∂dimE−1

��

HdimE−1∂C

∂dimE−1

��

β // H0(E−C)

HdimE(E−C) HdimE(E−C)∆

// H0(E−C),

The composite of the middle row is Alexander Duality by definition. Hence Hn−1(∂C ⊂C) is injective. Moreover, the sequence

Hn−1CHn−1(∂C⊂C)−−−−−−−−−→ Hn−1∂C

∂n−1

−−−→ Hnclo(C)|C−∂CConstC−∂C;R

is exact and its last term is trivial by the equalities

Hnclo(C)|C−∂CConstC−∂C;R = H

clo(C)|C−∂C0 ConstC−∂C;R = 0,

the first from Poincare Duality and the second from C − ∂C a filtered union of elements inclo(C)|(C−∂C) on which H0 is trivial [5, Exercise 26]. Then Hn−1(∂C ⊂ C), and hencealso α+, are isomorphisms.

The arrow β, whose composite with the isomorphism Hn−1(∂C ⊂ C) is an isomorphismAD : Hn−1C → H0(E−C), is an isomorphism. Moreover, β sends an orientation class on aconnected component represented by x to the path-component in E−C whose closure in Econtains x. Hence β∆−1ιC restricts and corestricts to the bijection

π0∂C ∼= π0(E−C),

and hence the isomorphism β+ in (8). Hence β+α+, a restriction and corestriction of AD,is an isomorphism.

Fix positive integer Q. Inductively assume the theorem holds for the case q = Q. Nowconsider the case q = Q+ 1.

Consider the following commutative diagram

(9) +Hn−1C //

��

##

∏x

+Hn−1Cx

∼=

&&

��Hn−1C

AD

##

∏xH

n−1(Cx⊂C)

// ∏xH

n−1Cx

&&

+Hq(E−C) //

��

∏x

+Hq−1(Ex − Cx)

��Hq(E−C) // ∏

xHq−1(Ex − Cx)


of solid arrows, where the front and back squares are pullback squares, the vertical arrows areinclusions, and the right arrow in the bottom square is defined by Alexander Dualities. Thereexists a dotted isomorphism making the side square commute by the inductive assumption.The bottom square in (9) commutes by a diagram chase and suitably compatible choices oforientations on E and its fibers. Hence there exists the desired dotted isomorphism in (9)by universal properties of pullbacks. �

The tools are now assembled to infer the complete criterion for evasion. The main theoremof this paper states that the positive cohomology +Hn−1C is the complete obstruction toevader capture – that an evasion path exists if and only if this cohomology is non-empty.

Corollary 4.2. For a coverage pair (E,C) over R with dimE > 2,

(10) +HdimE−2C 6= ∅

if and only if the restriction E−C → R of the projection E → R admits a section.

Proof. Observe that

+HdimE−2C 6= ∅ ⇐⇒ +H1(E−C) 6= ∅ Theorem 4.1

⇐⇒ E−C admits a section Lemmas 2.9, 2.8

�

5. Computation

We decompose the requisite calculation of positive cohomology in the criterion as a limitof local positive cohomology cones. Such a limit is naturally described as the global sectionsof a cellular sheaf. A cellular sheaf of cones on R is a functor from the poset of cells ina stratification of R, into vertices and edges, to Cones. We notate the global sections of acellular sheaf H of cones on R as ΓH; formally

ΓH = limcH(c),

where the limit, indexed over all vertices and edges of R, is taken in Cones. Cellular sheavesare well-suited to applications involving computation, such as the cellular complexes usedto compute ordinary homology; see, e.g., [8, 11, 20].

Example 5.1. For a fiberwise oriented space P over R equipped with a stratification,

+HnPst(−)

defines a cellular sheaf of cones on R, sending each cell c in R to the positive cohomology+HnPst(c) of the fiberwise oriented space Pst(c) over the open star of c in R and sending eachinclusion v 6 e of vertex into edge to an appropriate restriction map of cohomologies.

The criterion for evasion decomposes as the global sections of a cellular sheaf of localpositive cohomologies.

Corollary 5.2. For coverage pair (E,C) over R and collection O of open subsets of R,

+HdimE−2C⋃O = lim

U∈O

+HdimE−2CU .


Proof. There exist natural isomorphisms+Hn−1C ∼= +Hn−1(E−C) Theorem 4.1

∼= limU

+H1(EU−CU ) Lemma 2.7

∼= limU

+Hn−1CU , Theorem 4.1.

�

Formulated in the language of sheaves,+Hn−1C = Γ+Hn−1Cst(−).

Corollary 5.2 is the starting point for local-to-global calculations of the positive cohomo-logical criterion. This allows us to break down +Hn−1C into three calculations.

(1) the cohomology cellular sheaf Hn−1Cst(−)

(2) the positive cohomology cellular subsheaf +Hn−1Cst(−)

(3) the global sections Γ+Hn−1Cst(−)

The first calculation is purely classical. The second calculation involves checking whichlocal sections in Hn−1(U) restrict to orientation classes on connected components of ∂Ctfor all open stars U in a stratification of R and all t ∈ U with Ct a connected manifold-with-compact-boundary. The third calculation is nontrivial. In this section, we show howto reduce it to a simple linear program.

Proposition 5.3. For a cellular sheaf H of cones on R equipped with a stratification,

(11) ΓH =∏v

H(v)∩ ker

(∏v

H(v)±(φv)v 7→(H±(e−6e)(φe− )−H±(e+6e)(φe+

))e−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→

∏e

H(e)±

),

where v ranges over all vertices and e ranges over all precompact edges in R, and e−, e+

denote the vertices of a precompact edge in R having boundaries e− < e+.

Otherwise said, global sections (H0) consist of positive local sections (data over v) whichagree when restricted to data over incident edges, using the kernel to measure agreement.This is a linear-algebraic view of why the evasion criterion of this paper works whereasothers do not: the positive cohomology sheaf +Hn−1C, instead of computing an Abeliankernel (as in ordinary cohomology), computes the positive kernel of a difference of two mapson positive cones.

This prompts an algorithmic approach to computation in the cellular setting. Let H be asheaf of positive, polyhedral cones in finite dimensional real vector spaces and maps betweensuch cones. Fixing positive bases for the stalks of H± over vertices and edges, the map

(12)∏v

H±(v)→∏e

H±(e),

in (11) yields a real matrix (a cellular coboundary matrix). The entries of this matrix aredifferences, and thus may be positive or negative. However, to contribute to H0, the kernelmust have nontrivial intersection with

∏vH(v), meaning that one must compute the kernel

subject to a sequence of inequalities.Computing ΓH is therefore reduced to the question of whether a known subspace of a

real vector space (the kernel of (12)) intersects a positive cone in that vector space. Bynormalizing the cone to, say, unit `1 distance to the origin, finding a nonzero element ofΓH is expressible as a simple linear programming problem. Assuming a fixed bound onthe complexity of the cones (the number of constraints), the solution to this problem via


standard methods is linear in the number of variables [17], in this case, the sum of thenumber of generators of H(vi) over all i.

6. Planar examples

We give a computation of the criterion for examples in the planar case (n = 2). The twoprincipal examples from [1] alluded to in §1 are the simplest examples with which to showhow +H1 determines evasion. Consider the coverage regions C over R illustrated in Figures5 and 6. Stratify R as shown in a manner compatible with the projection map of C to R:this decomposition has four vertices {vi}41 and five edges {ej}51 with e1 and e5 extending to±∞.

v1

v2

e2

e1

e3

e4

e5v

4v3

Figure 5. A simple planar case in which an evasion path exists. Shownis a parameterized space of (unbounded, shaded) sensed regions, Ct and(bounded) evasion sets as a function of time.

v1

v2

e2

e1

e3

e4

e5v

4v3

Figure 6. The analogue of Figure 5 for which no evasion path exists.

With this subdivision of R, the associated sheaves are cellular, with stalks computed interms of local sections. (Specifically, the stalk over a vertex vi equals the value of the sheafon the open star st(vi).) In both cases, the dimension of the stalks of +H1 are given asfollows:


• There is one generator over e1 and e5; two over e2 and e4; and three over e3.• There is one generator over v1 and v4; and three over v2 and v3.

Label all these generators by the cells in the time axis, with superscripts where ”t” connotesa generator associated to the top hole, ”b” the bottom, and ”m” the middle. In all cases,the cones determined by +H1 are positive orthants in Rk for k the number of generators ofthe stalk.

All the induced maps from stalks over vi to stalks over ei and ei+1 are either trivialinclusions or projections given by correspondence of generators, and it is in this regardthat the sheaves for Figures 5 and 6 differ. The global criterion Γ +H1 is computed viaEquation (11) using these restriction maps with the appropriate signs. Note that, since weare computing cellular cohomology, we must zero-out all the maps to e1 and e5, since thesehave noncompact closure in R, as per the discussion above. We present this data in theform of a matrix with entries of the form a+

ij − a−ij , so as to see the explicit dependence on

whether the term comes from the left or right in time. For the system of Figure 5, thismatrix is:

(13)∏v

+H1(δ+v )−+H1(δ−v ) =

vt1 vt2 vm2 vb2 vt3 vm3 vb3 vb4et2 0−1 1−0 1−0 0−0 0−0 0−0 0−0 0−0eb2 0−0 0−0 0−0 1−0 0−0 0−0 0−0 0−0et3 0−0 0−1 0−0 0−0 1−0 0−0 0−0 0−0em3 0−0 0−0 0−1 0−0 0−0 1−0 0−0 0−0eb3 0−0 0−0 0−0 0−1 0−0 0−0 1−0 0−0et4 0−0 0−0 0−0 0−0 0−1 0−0 0−0 0−0eb4 0−0 0−0 0−0 0−0 0−0 0−1 0−1 1−0

One computes that +H1C 6= ∅ as follows. The kernel of (13) is one-dimensional with apositive generator given by the sum of generators vt1 + vm2 + vm3 + vb4. These generators, ofcourse, correspond to the dual evasion path sequence easily observed from Figure 5. The co-homology +H1C is thus isomorphic to an open ray. In this example, one can observe directlythat the kernel intersects the positive orthant, and no linear programming is necessary. Alarger, more opaque example would lead to more complex computation.

For Figure 6, the positive-negative coboundary matrix on the sheaf +H1 is:

(14)∏v

+H1(δ+v )−+H1(δ−v ) =

vt1 vt2 vm2 vb2 vt3 vm3 vb3 vb4et2 0−1 1−0 0−0 0−0 0−0 0−0 0−0 0−0eb2 0−0 0−0 1−0 1−0 0−0 0−0 0−0 0−0et3 0−0 0−1 0−0 0−0 1−0 0−0 0−0 0−0em3 0−0 0−0 0−1 0−0 0−0 1−0 0−0 0−0eb3 0−0 0−0 0−0 0−1 0−0 0−0 1−0 0−0et4 0−0 0−0 0−0 0−0 0−1 0−1 0−0 0−0eb4 0−0 0−0 0−0 0−0 0−0 0−0 0−1 1−0

One computes that this matrix has, as before, one-dimensional kernel; however, thegenerator of this kernel is

vt1 + vt2 + vt3 − vm3 − vm2 + vb2 + vb3 + v4b ,

which has both positive and negative terms. The intersection of this kernel with the openpositive orthant is empty; therefore, +H1C = ∅, and there is no evasion path.


7. Concluding remarks

(1) For genuine applications to pursuit problems, much of the structure here introducedis adaptable. For example, the presence of obstacles in the free space (an importantconsideration in practice) is admissible by including the obstacle regions within thecoverage domain. The use of a sensor network to determine the coverage region(by means of, say, a Vietoris-Rips complex) may be likewise permissible, if therelevant cohomology and orientation data are discernible. This is an interestingopen problem.

(2) This work has considered only the connectivity data about the evasion paths. Thefull space of evasion paths may have interesting topological features beyond connec-tivity alone. There would appear also to be virtue in augmenting π0 with additionaldata about evaders: number, identity, team-membership, or other features, much inthe same way that feature vectors are used in target tracking. It is to be suspectedthat sheaf constructions permit a great deal of such information to be encoded.

(3) The use of linear programming to compute the positive cohomology in this paper isperhaps the beginnings of a broader set of techniques in computational equalizersand what we would call homological programming.

(4) Finally, the utility of directed co/homology appears to be relevant to other applica-tions in which directedness is crucial, including Bayesian belief networks.

Appendix A. Existence of sections

We give homotopical criteria for open subspaces of vector bundles over R to admit sec-tions.

Lemma A.1. Consider the following data.

(1) a CW structure on R; and(2) an open subspace P of a Euclidean space over R

The natural function secP (R) → limc π0secP (c), where c denotes a closed cell in R, issurjective.

Proof. Fix a natural number n. We take P to be an open subspace of Rn+1 such that τPis the projection onto the last coordinate. Let ξ be projection P → Rn onto the first ncoordinates. We take the vertices of R to be the integers without loss of generality.

Let i denote an integer, and consider si ∈ secP ([i, i+ 1]) for each i and path γi : si−1(i) ;si(i) for each i. It suffices to construct s ∈ secP (R) with s|(i,i+1) ∼ si for each i.

Fix i.The subspace im γi ⊂ P admits an open cover O consisting of open hyperrectangles

because Rn, and hence also its open subspace P , admit open bases consisting of openhyperrectangles. We can take O to be finite by [i, i + 1] and hence im γi compact. Hence⋂U∈O τP (U) is an open neighborhood in R of i, and hence contains a subset of the form

[i, i+ 2εi] for 0 < εi < 1/2. Hence

P ⊃⋃

O ⊃⋃V ∈O

ξ(V )× τP (V ) ⊃⋃V ∈O

ξ(V )×⋂U∈O

τPU ⊃ ξ(im γi)× [i, i+ 2εi]


Hence we can define a section s : R→ P to P by

s(x) =

(ξ(γi((x−i)/εi), x), x ∈ [i, i+ εi]

(ξ(si(2x− 2εi − i)), x), x ∈ [i+ εi, i+ 2εi]

si(x) x ∈ [i+ 2εi, i+ 1]

For each i ∈ Z, s|[i,i+1] ∼ si along the homotopy hi defined by

hi(x, t) =

(ξ(γi(1− t+ (x−i)/εi), x), x ∈ [i, i+ tεi]

(ξ(si(2x− 2tεi − i)), x), x ∈ [i+ tεi, i+ 2tεi]

si(x) x ∈ [i+ 2tεi, i+ 1]

�

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Departments of Mathematics & Electrical/Systems Engineering, Univ. PennsylvaniaE-mail address: [email protected]

Department of Mathematics, Ohio State UniversityE-mail address: [email protected]

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Introduction E;Cghrist/preprints/pursuit...Theorem 4.1. For coverage pair (E;C) over Rq with dimE>2 and 0 6q6dimE 2, +HdimE q 1C˘=+H q(E C) Along the way, we investigate properties